Focusing and sensing apparatus, methods, and systems

ABSTRACT

Apparatus, methods, and systems provide focusing, focus-adjusting, and sensing. In some approaches the focus-adjusting includes providing an extended depth of focus greater than a nominal depth of focus. In some approaches the focus-adjusting includes focus-adjusting with a transformation medium, where the transformation medium may include an artificially-structured material such as a metamaterial.

RELATED APPLICATIONS

For purposes of the USPTO extra-statutory requirements, the presentapplication constitutes a continuation-in-part of U.S. patentapplication Ser. No. 12/156,443, entitled FOCUSING AND SENSINGAPPARATUS, METHODS, AND SYSTEMS, naming Jeffrey A. Bowers, Roderick A.Hyde, Edward K. Y. Jung, John Brian Pendry, David Schurig, David R.Smith, Clarence T. Tegreene, Thomas A. Weaver, Charles Whitmer, andLowell L. Wood, Jr. as inventors, filed May 30, 2008, which is currentlyco-pending, or is an application of which a currently co-pendingapplication is entitled to the benefit of the filing date.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is related to and claims the benefit of theearliest available effective filing date(s) from the following listedapplication(s) (the “Related Applications”) (e.g., claims earliestavailable priority dates for other than provisional patent applicationsor claims benefits under 35 USC 119(e) for provisional patentapplications, for any and all parent, grandparent, great-grandparent,etc. applications of the Related Application(s)).

The United States Patent Office (USPTO) has published a notice to theeffect that the USPTO's computer programs require that patent applicantsreference both a serial number and indicate whether an application is acontinuation or continuation-in-part. Stephen C. Kunin, Benefit ofPrior-Filed Application, USPTO Official Gazette Mar. 18, 2003, availableat http://www.uspto.gov/web/offices/com/sol/og/2003/week 11/patbene.htm.The present Applicant Entity (hereinafter “Applicant”) has providedabove a specific reference to the application(s) from which priority isbeing claimed as recited by statute. Applicant understands that thestatute is unambiguous in its specific reference language and does notrequire either a serial number or any characterization, such as“continuation” or “continuation-in-part,” for claiming priority to U.S.patent applications. Notwithstanding the foregoing, Applicantunderstands that the USPTO's computer programs have certain data entryrequirements, and hence Applicant is designating the present applicationas a continuation-in-part of its parent applications as set forth above,but expressly points out that such designations are not to be construedin any way as any type of commentary and/or admission as to whether ornot the present application contains any new matter in addition to thematter of its parent application(s).

All subject matter of the Related Applications and of any and allparent, grandparent, great-grandparent, etc. applications of the RelatedApplications is incorporated herein by reference to the extent suchsubject matter is not inconsistent herewith.

TECHNICAL FIELD

The application discloses apparatus, methods, and systems that mayrelate to electromagnetic responses that include focusing,focus-adjusting, and sensing.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 depicts a first configuration of a focusing structure and afocus-adjusting structure.

FIG. 2 depicts a first coordinate transformation.

FIG. 3 depicts a second configuration of a focusing structure and afocus-adjusting structure.

FIG. 4 depicts a second coordinate transformation.

FIG. 5 depicts a focusing structure and a focus-adjusting structure witha spatial separation.

FIG. 6 depicts a focusing structure and a focus-adjusting structurewithout a spatial separation.

FIG. 7 depicts a first process flow.

FIG. 8 depicts a second process flow.

FIG. 9 depicts a system that includes a focus-adjusting unit and acontroller.

DETAILED DESCRIPTION

In the following detailed description, reference is made to theaccompanying drawings, which form a part hereof. In the drawings,similar symbols typically identify similar components, unless contextdictates otherwise. The illustrative embodiments described in thedetailed description, drawings, and claims are not meant to be limiting.Other embodiments may be utilized, and other changes may be made,without departing from the spirit or scope of the subject matterpresented here.

Transformation optics is an emerging field of electromagneticengineering. Transformation optics devices include lenses that refractelectromagnetic waves, where the refraction imitates the bending oflight in a curved coordinate space (a “transformation” of a flatcoordinate space), e.g. as described in A. J. Ward and J. B. Pendry,“Refraction and geometry in Maxwell's equations,” J. Mod. Optics 43, 773(1996), J. B. Pendry and S. A. Ramakrishna, “Focusing light usingnegative refraction,” J. Phys. [Cond. Matt.] 15, 6345 (2003), D. Schuriget al, “Calculation of material properties and ray tracing intransformation media,” Optics Express 14, 9794 (2006) (“D. Schurig et al(1)”), and in U. Leonhardt and T. G. Philbin, “General relativity inelectrical engineering,” New J. Phys. 8, 247 (2006), each of which isherein incorporated by reference. The use of the term “optics” does notimply any limitation with regards to wavelength; a transformation opticsdevice may be operable in wavelength bands that range from radiowavelengths to visible wavelengths.

A first exemplary transformation optics device is the electromagneticcloak that was described, simulated, and implemented, respectively, inJ. B. Pendry et al, “Controlling electromagnetic waves,” Science 312,1780.(2006); S. A. Cummer et al, “Full-wave simulations ofelectromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006);and D. Schurig et al, “Metamaterial electromagnetic cloak at microwavefrequencies,” Science 314, 977 (2006) (“D. Schurig et al (2)”); each ofwhich is herein incorporated by reference. See also J. Pendry et al,“Electromagnetic cloaking method,” U.S. patent application Ser. No.11/459,728, herein incorporated by reference. For the electromagneticcloak, the curved coordinate space is a transformation of a flat spacethat has been punctured and stretched to create a hole (the cloakedregion), and this transformation corresponds to a set of constitutiveparameters (electric permittivity and magnetic permeability) for atransformation medium wherein electromagnetic weaves are refractedaround the hole in imitation of the curved coordinate space.

A second exemplary transformation optics device is illustrated byembodiments of the electromagnetic compression structure described in J.B. Pendry, D. Schurig, and D. R. Smith, “Electromagnetic compressionapparatus, methods, and systems,” U.S. patent application Ser. No.11/982,353; and in J. B. Pendry, D. Schurig, and D. R. Smith,“Electromagnetic compression apparatus, methods, and systems,” U.S.patent application Ser. No. 12/069,170; each of which is hereinincorporated by reference. In embodiments described therein, anelectromagnetic compression structure includes a transformation mediumwith constitutive parameters corresponding to a coordinatetransformation that compresses a region of space intermediate first andsecond spatial locations, the effective spatial compression beingapplied along an axis joining the first and second spatial locations.The electromagnetic compression structure thereby provides an effectiveelectromagnetic distance between the first and second spatial locationsgreater than a physical distance between the first and second spatiallocations.

A third exemplary transform optics device is illustrated by embodimentsof the electromagnetic cloaking and/or translation structure describedin J. T. Kare, “Electromagnetic cloaking apparatus, methods, andsystems,” U.S. patent application Ser. No. 12/074,247; and in J. T.Kare, “Electromagnetic cloaking apparatus, methods, and systems,” U.S.patent application Ser. No. 12/074,248; each of which is hereinincorporated by reference. In embodiments described therein, anelectromagnetic translation structure includes a transformation mediumthat provides an apparent location of an electromagnetic transducerdifferent then an actual location of the electromagnetic transducer,where the transformation medium has constitutive parameterscorresponding to a coordinate transformation that maps the actuallocation to the apparent location. Alternatively or additionally,embodiments include an electromagnetic cloaking structure operable todivert electromagnetic radiation around an obstruction in a field ofregard of the transducer (and the obstruction can be anothertransducer).

Additional exemplary transformation optics devices are described in D.Schurig et al, “Transformation-designed optical elements,” Opt. Exp. 15,14772 (2007); M. Rahm et al, “Optical design of reflectionless complexmedia by finite embedded coordinate transformations,” Phys. Rev. Lett.100, 063903 (2008); and A. Kildishev and V. Shalaev, “Engineering spacefor light via transformation optics,” Opt. Lett. 33, 43 (2008); each ofwhich is herein incorporated by reference.

In general, for a selected coordinate transformation, a transformationmedium can be identified wherein electromagnetic waves refract as ifpropagating in a curved coordinate space corresponding to the selectedcoordinate transformation. Constitutive parameters of the transformationmedium can be obtained from the equations:{tilde over (∈)}^(i′j′) =|det(Λ)|⁻¹Λ_(i) ^(i′)Λ_(j) ^(j′)∈^(ij)  (1){tilde over (μ)}^(i′j′) =|det(Λ)|⁻¹Λ_(i) ^(i′)Λ_(j) ^(j′)μ^(ij)  (2)where {tilde over (∈)} and {tilde over (μ)} are the permittivity andpermeability tensors of the transformation medium, ∈ and μ are thepermittivity and permeability tensors of the original medium in theuntransformed coordinate space, and

$\begin{matrix}{\Lambda_{i}^{i^{\prime}} = \frac{\partial x^{i^{\prime}}}{\partial x^{i}}} & (3)\end{matrix}$is the Jacobian matrix corresponding to the coordinate transformation.In some applications, the coordinate transformation is a one-to-onemapping of locations in the untransformed coordinate space to locationsin the transformed coordinate space, and in other applications thecoordinate transformation is a many-to-one mapping of locations in theuntransformed coordinate space to locations in the transformedcoordinate space. Some coordinate transformations, such as many-to-onemappings, may correspond to a transformation medium having a negativeindex of refraction. In some applications, only selected matrix elementsof the permittivity and permeability tensors need satisfy equations (1)and (2), e.g. where the transformation optics response is for a selectedpolarization only. In other applications, a first set of permittivityand permeability matrix elements satisfy equations (1) and (2) with afirst Jacobian Λ, corresponding to a first transformation opticsresponse for a first polarization of electromagnetic waves, and a secondset of permittivity and permeability matrix elements, orthogonal (orotherwise complementary) to the first set of matrix elements, satisfyequations (1) and (9) with a second Jacobian Λ′, corresponding to asecond transformation optics response for a second polarization ofelectromagnetic waves. In yet other applications, reduced parameters areused that may not satisfy equations (1) and (2), but preserve productsof selected elements in (1) and selected elements in (2), thuspreserving dispersion relations inside the transformation medium (see,for example, D. Schurig et al (2), supra, and W. Cai et al, “Opticalcloaking with metamaterials,” Nature Photonics 1, 224 (2007), hereinincorporated by reference). Reduced parameters can be used, for example,to substitute a magnetic response for an electric response, or viceversa. While reduced parameters preserve dispersion relations inside thetransformation medium (so that the ray or wave trajectories inside thetransformation medium are unchanged from those of equations (1) and(2)), they may not preserve impedance characteristics of thetransformation medium, so that rays or waves incident upon a boundary orinterface of the transformation medium may sustain reflections (whereasin general a transformation medium according to equations (1) and (2) issubstantially nonreflective). The reflective or scatteringcharacteristics of a transformation medium with reduced parameters canbe substantially reduced or eliminated by a suitable choice ofcoordinate transformation, e.g. by selecting a coordinate transformationfor which the corresponding Jacobian Λ (or a subset of elements thereof)is continuous or substantially continuous at a boundary or interface ofthe transformation medium (see, for example, W. Cai et al, “Nonmagneticcloak with minimized scattering,” Appl. Phys. Lett. 91, 111105 (2007),herein incorporated by reference).

In general, constitutive parameters (such as permittivity andpermeability) of a medium responsive to an electromagnetic wave can varywith respect to a frequency of the electromagnetic wave (orequivalently, with respect to a wavelength of the electromagnetic wavein vacuum or in a reference medium). Thus, a medium can haveconstitutive parameters ∈₁, μ₁, etc. at a first frequency, andconstitutive parameters ∈₂, μ₂, etc. at a second frequency; and so onfor a plurality of constitutive parameters at a plurality offrequencies. In the context of a transformation medium, constitutiveparameters at a first frequency can provide a first response toelectromagnetic waves at the first frequency, corresponding to a firstselected coordinate transformation, and constitutive parameters at asecond frequency can provide a second response to electromagnetic wavesat the second frequency, corresponding to a second selected coordinatetransformation; and so on: a plurality of constitutive parameters at aplurality of frequencies can provide a plurality of responses toelectromagnetic waves corresponding to a plurality of coordinatetransformations. In some embodiments the first response at the firstfrequency is substantially nonzero (i.e. one or both of ∈₁ and μ₁ issubstantially non-unity), corresponding to a nontrivial coordinatetransformation, and a second response at a second frequency issubstantially zero (i.e. ∈₂ and μ₂ are substantially unity),corresponding to a trivial coordinate transformation (i.e. a coordinatetransformation that leaves the coordinates unchanged); thus,electromagnetic waves at the first frequency are refracted(substantially according to the nontrivial coordinate transformation),and electromagnetic waves at the second frequency are substantiallynonrefracted. Constitutive parameters of a medium can also change withtime (e.g. in response to an external input or control signal), so thatthe response to an electromagnetic wave can vary with respect tofrequency and/or time. Some embodiments may exploit this variation withfrequency and/or time to provide respective frequency and/or timemultiplexing/demultiplexing of electromagnetic waves. Thus, for example,a transformation medium can have a first response at a frequency at timet₁, corresponding to a first selected coordinate transformation, and asecond response at the same frequency at time t₂, corresponding to asecond selected coordinate transformation. As another example, atransformation medium can have a response at a first frequency at timet₁, corresponding to a selected coordinate transformation, andsubstantially the same response at a second frequency at time t₂. In yetanother example, a transformation medium can have, at time t₁, a firstresponse at a first frequency and a second response at a secondfrequency, whereas at time t₂, the responses are switched, i.e. thesecond response (or a substantial equivalent thereof) is at the firstfrequency and the first response (or a substantial equivalent thereof)is at the second frequency. The second response can be a zero orsubstantially zero response. Other embodiments that utilize frequencyand/or time dependence of the transformation medium will be apparent toone of skill in the art.

Constitutive parameters such as those of equations (1) and (2) (orreduced parameters derived therefrom) can be realized usingartificially-structured materials. Generally speaking, theelectromagnetic properties of artificially-structured materials derivefrom their structural configurations, rather than or in addition totheir material composition.

In some embodiments, the artificially-structured materials are photoniccrystals. Some exemplary photonic crystals are described in J. D.Joannopoulos et al, Photonic Crystals: Molding the Flow of Light, 2^(nd)Edition, Princeton Univ. Press, 2008, herein incorporated by reference.In a photonic crystals, photonic dispersion relations and/or photonicband gaps are engineered by imposing a spatially-varying pattern on anelectromagnetic material (e.g. a conducting, magnetic, or dielectricmaterial) or a combination of electromagnetic materials. The photonicdispersion relations translate to effective constitutive parameters(e.g. permittivity, permeability, index of refraction) for the photoniccrystal. The spatially-varying pattern is typically periodic,quasi-periodic, or colloidal periodic, with a length scale comparable toan operating wavelength of the photonic crystal.

In other embodiments, the artificially-structured materials aremetamaterials. Some exemplary metamaterials are described in R. A. Hydeet al, “Variable metamaterial apparatus,” U.S. patent application Ser.No. 11/355,493; D. Smith et al, “Metamaterials,” InternationalApplication No. PCT/US2005/026052; D. Smith et al, “Metamaterials andnegative refractive index,” Science 305, 788 (2004); D. Smith et al,“Indefinite materials,” U.S. patent application Ser. No. 10/525,191; C.Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission LineTheory and Microwave Applications, Wiley-Interscience, 2006; N. Enghetaand R. W. Ziolkowski, eds., Metamaterials: Physics and EngineeringExplorations, Wiley-Interscience, 2006; and A. K. Sarychev and V. M.Shalaev, Electrodynamics of Metamaterials, World Scientific, 2007; eachof which is herein incorporated by reference.

Metamaterials generally feature subwavelength elements, i.e. structuralelements with portions having electromagnetic length scales smaller thanan operating wavelength of the metamaterial, and the subwavelengthelements haste a collective response to electromagnetic radiation thatcorresponds to an effective continuous medium response, characterized byan effective permittivity, an effective permeability, an effectivemagnetoelectric coefficient, or any combination thereof. For example,the electromagnetic radiation may induce charges and/or currents in thesubwavelength elements, whereby the subwavelength elements acquirenonzero electric and/or magnetic dipole moments. Where the electriccomponent of the electromagnetic radiation induces electric dipolemoments, the metamaterial has an effective permittivity; where themagnetic component of the electromagnetic radiation induces magneticdipole moments, the metamaterial has an effective permeability; andwhere the electric (magnetic) component induces magnetic (electric)dipole moments (as in a chiral metamaterial), the metamaterial has aneffective magnetoelectric coefficient. Some metamaterials provide anartificial magnetic response; for example, split-ring resonators(SRRs)—or other LC or plasmonic resonators—built from nonmagneticconductors can exhibit an effective magnetic permeability (c.f. J. B.Pendry et al, “Magnetism from conductors and enhanced nonlinearphenomena,” IEEE Trans. Micro. Theo. Tech. 47, 2075 (1999), hereinincorporated by reference). Some metamaterials have “hybrid”electromagnetic properties that emerge partially from structuralcharacteristics of the metamaterial, and partially from intrinsicproperties of the constituent materials. For example, G. Dewar, “A thinwire array and magnetic host structure with n<0,” J. Appl. Phys. 97,10Q101 (2005), herein incorporated by reference, describes ametamaterial consisting of a wire array (exhibiting a negativepermeability as a consequence of its structure) embedded in anonconducting ferrimagnetic host medium (exhibiting an intrinsicnegative permeability). Metamaterials can be designed and fabricated toexhibit selected permittivities, permeabilities, and/or magnetoelectriccoefficients that depend upon material properties of the constituentmaterials as well as shapes, chiralities, configurations, positions,orientations, and couplings between the subwavelength elements. Theselected permittivites, permeabilities, and/or magnetoelectriccoefficients can be positive or negative, complex (having loss or gain),anisotropic, variable in space (as in a gradient index lens), variablein time (e.g. in response to an external or feedback signal), variablein frequency (e.g. in the vicinity of a resonant frequency of themetamaterial), or any combination thereof. The selected electromagneticproperties can be provided at wavelengths that range from radiowavelengths to infrared/visible wavelengths; the latter wavelengths areattainable, e.g., with nanostructured materials such as nanorod pairs ornano-fishnet structures (c.f. S. Linden et al, “Photonic metamaterials:Magnetism at optical frequencies,” IEEE J. Select. Top. Quant. Elect.12, 1097 (2006) and V. Shalaev, “Optical negative-index metamaterials,”Nature Photonics 1, 41 (2007), both herein incorporated by reference).An example of a three-dimensional metamaterial at optical frequencies,an elongated-split-ring “woodpile” structure, is described in M. S. Rillet al, “Photonic metamaterials by direct laser writing and silverchemical vapour deposition,” Nature Materials advance onlinepublication, May 11, 2008, (doi: 10.1038/nmat2197).

While many exemplary metamaterials are described as including discreteelements, some implementations of metamaterials may include non-discreteelements or structures. For example, a metamaterial may include elementscomprised of sub-elements, where the sub-elements are discretestructures (such as split-ring resonators, etc.), or the metamaterialmay include elements that are inclusions, exclusions, layers, or othervariations along some continuous structure (e.g. etchings on asubstrate). Some examples of layered metamaterials include: a structureconsisting of alternating doped/intrinsic semiconductor layers (cf. A.J. Hoffman, “Negative refraction in semiconductor metamaterials,” NatureMaterials 6, 946 (2007), herein incorporated by reference), and astructure consisting of alternating metal/dielectric layers (cf. A.Salandrino and N. Engheta, “Far-field subdiffraction optical microscopyusing metamaterial crystals: Theory and simulations,” Phys. Rev. B 74,075103 (2006); and Z. Jacob et al, “Optical hyperlens: Far-field imagingbeyond the diffraction limit,” Opt. Exp. 14, 8247 (2006); each of whichis herein incorporated by reference). The metamaterial mats includeextended structures having distributed electromagnetic responses (suchas distributed inductive responses, distributed capacitive responses,and distributed inductive-capacitive responses). Examples includestructures consisting of loaded and/or interconnected transmission lines(such as microstrips and striplines), artificial ground plane structures(such as artificial perfect magnetic conductor (PMC) surfaces andelectromagnetic band gap (EGB) surfaces), and interconnected/extendednanostructures (nano-fishnets, elongated SRR woodpiles, etc.).

With reference now to FIG. 1, an illustrative embodiment is depictedthat includes a focusing structure 110 and a focus-adjusting structure120. This and other drawings, unless context dictates otherwise, canrepresent a planar view of a three-dimensional embodiment, or atwo-dimensional embodiment (e.g. in FIG. 1 where the transducers arepositioned inside a metallic or dielectric slab waveguide orientednormal to the page). The focusing structure receives inputelectromagnetic energy, depicted as solid rays 102; in this example, theinput electromagnetic energy radiates from an electromagnetic source 101positioned on an optical axis 112 of the focusing structure (the use ofa ray description, in FIG. 1 and elsewhere, is a heuristic conveniencefor purposes of visual illustration, and is not intended to connote anylimitations or assumptions of geometrical optics; further, the elementsdepicted in FIG. 1 can have spatial dimensions that are variously lessthan, greater than, or comparable to a wavelength of interest). Thesolid rays 103 represent output electromagnetic energy from the focusingstructure. This output electromagnetic energy is received by thefocus-adjusting structure 120, which influences the propagation of theoutput electromagnetic energy (e.g. by refracting the rays 103, asdepicted). The dashed rays 104 represent the “nominal” propagation ofthe output electromagnetic energy, i.e. the propagation that would occurin the absence of the focus-adjusting structure. As the dashed raysindicate, the focusing structure 110 provides a nominal focusing region130 having a nominal depth of focus 132; in this example, the nominalfocusing region 130 is depicted as a slab having a thickness equal tothe nominal depth of focus 132 and centered on a nominal focal plane 134(where the dashed rays converge). The focus-adjusting structure 120influences the propagation of the output electromagnetic energy toprovide an actual focusing region 140 different than the nominalfocusing region 130, the actual focusing region having an actual depthof focus 142 which is an extended depth of focus greater than thenominal depth of focus; in this example, the actual focusing region 140is depicted as a slab having a thickness equal to the actual depth offocus 142 and centered on an actual focal plane 144. Embodimentsoptionally include one or more electromagnetic sensors (schematicallydepicted as ellipses 150) positioned within the actual focusing region.The dashed ellipse 152 represents a nominal sensor positioning, i.e. asensor positioning within the nominal focusing region 130 that couldapply in the absence of the focus-adjusting structure (in the presentexample, sensors are positioned only along the optical axis 112 of thefocusing structure, but this is not intended to be limiting). As thefigure demonstrates, the actual depth of focus may, in some embodiments,accommodate more sensors than the nominal depth of focus.

The focusing structure 110 is depicted in FIG. 1 as having a lens-likeshape, but this is a schematic illustration and is not intended to belimiting. In various embodiments, focusing structures can includereflective structures (e.g. parabolic dish reflectors), refractivestructures (e.g. dielectric, microwave, gradient index, or metamateriallenses), diffractive structures (e.g. Fresnel zone plates), antennastructures (e.g. antenna director elements, antenna arrays), waveguidingstructures (e.g. waveguides, transmission lines, and coherent bundlesthereof) and various combinations, assemblies, portions, and hybridsthereof (such as an optical assembly or a refractive-diffractive lens).In general, a focusing structure provides a nominal focusing regionwherein electromagnetic energy coupled to the focusing structure isnominally substantially concentrated (i.e. wherein electromagneticenergy coupled to the focusing structure is substantially concentratedin the absence of a focus-adjusting structure; the influence of thefocus-adjusting structure is discussed below). In some embodiments, thenominal focusing region may be a planar or substantially planar slab(e.g. 130 in FIG. 1) having a thickness corresponding to a nominal depthof focus of the focusing structure (e.g. 132 in FIG. 1) and centered ona nominal focal plane (e.g. 134 in FIG. 1). In other embodiments, thenominal focusing region may be a non-planar region, e.g. acylindrically-, spherically-, ellipsoidally-, or otherwise-curved slabhaving a thickness corresponding to a nominal depth of focus of thefocusing structure, and the non-planar region may enclose a non-planarfocal surface (such as a Petzval, sagittal, or transverse focalsurface). In some embodiments the focusing structure defines an opticalaxis (such as that depicted as element 112 in FIG. 1) as a symmetry orcentral axis of the focusing structure, and the optical axis provides anaxial direction (such as the axial unit vector 160 in FIG. 1), withtransverse directions defined perpendicular thereto (such as thetransverse unit vectors 161 and 162 in FIG. 1). More generally, one maydefine an axial direction corresponding to the nominal depth of focus(with transverse directions perpendicular thereto), so that the nominaldepth of focus is equivalent to a nominal axial dimension of the nominalfocusing region. This is consistent with FIG. 1, where the nominalfocusing region is a planar slab, and the axial direction corresponds toa unit vector normal to the slab. Where the nominal focusing region iscurved, the axial direction can vary along the transverse extent of thefocusing region. For example, where the nominal focusing region is acylindrically- or spherically-curved slab, the axial directioncorresponds to a radius unit vector (and the transverse directionscorrespond to height/azimuth unit vectors or azimuth/zenith unitvectors, respectively); where the nominal focusing region is anotherwise-curved slab, the axial direction corresponds to a vectorlocally normal to the slab surface (and the transverse directionscorrespond to orthogonal unit vectors locally tangent to the slabsurface).

In some embodiments the nominal depth of focus of a focusing structuremay be related to an f-number f/# of the focusing structure, and/or to aresolution length l in a transverse direction of the nominal focusingregion. The f-number can correspond to a ratio of focal length toaperture diameter for the focusing structure, and may (inversely)characterize the convergence of electromagnetic energy towards thenominal focusing region; moreover, the convergence can correspond to aratio of nominal depth of focus to transverse resolution length; so thefollowing general relation may apply for the focusing structure:

$\begin{matrix}{f/{\left. \# \right.\sim\;\frac{d_{N}}{\ell}}} & (4)\end{matrix}$where d_(N) is the nominal depth of focus. Thus, for a fixed f-number, alarger (smaller) nominal depth of focus corresponds to a larger(smaller) resolution length. This is demonstrated, for example, in FIG.1, which indicates a resolution length 136 corresponding to a transverseextent of the nominal rays 104 at the surface of the nominal focusingregion 130. In some embodiments the resolution length may corresponds toa circle of confusion (CoC) diameter limit for image blur perception,and/or the resolution length may correspond to a transverse extent of(the sensitive area of) an individual sensor or multiplet of sensors, asfurther discussed below.

In the context of a focus-adjusting structure (e.g. 120 in FIG. 1), onemay further define (in addition to a nominal focusing region discussedabove) an actual focusing region as a region wherein electromagneticenergy coupled to the focusing structure is substantially concentratedin the presence of the focus-adjusting structure. In some embodiments,the actual focusing region may be a planar or substantially planar slab(e.g. 140 in FIG. 1) having a thickness corresponding to an actual depthof focus of the focusing structure (e.g. 142 in FIG. 1) and centered onan actual focal plane (e.g. 144 in FIG. 1). In other embodiments, theactual focusing region may be a non-planar region, e.g. acylindrically-, spherically-, ellipsoidally-, or otherwise-curved slabhaving a thickness corresponding to an actual depth of focus of thefocusing structure, and the non-planar region may enclose a non-planarfocal surface (such as a Petzval, sagittal, or transverse focalsurface). In embodiments where the nominal focusing region and theactual focusing region are substantially parallel (for substantiallyplanar slabs), substantially concentric/confocal (for substantiallycylindrically-, spherically, or ellipsoidally-curved slabs), orotherwise substantially co-curved, the axial and transverse directions(defined previously for the nominal focusing region) also apply to thegeometry of the actual focusing region; i.e. the axial directioncorresponds to both the nominal and axial depths of focus (withtransverse directions perpendicular thereto), and the actual depth offocus is equivalent to a actual axial dimension of the actual focusingregion as measured along the axial dimension defined previously.

In some embodiments a focus-adjusting structure, such as that depictedin FIG. 1, includes a transformation medium. For example, the raytrajectories 103 in FIG. 1 correspond to a coordinate transformationthat is a uniform spatial dilation along the axial direction 160 (withinthe axial extent of the focus-adjusting structure 120); this coordinatetransformation can be used to identify constitutive parameters for acorresponding transformation medium (e.g. as provided in equations (1)and (2), or reduced parameters obtained therefrom) that responds toelectromagnetic radiation as in FIG. 1. Explicitly, for the example ofFIG. 1, defining as an untransformed (nominal) focal distance and z′ asa transformed (actual) focal distance (where the distances are measuredfrom the rear vertex 170 along the axial direction 160), the coordinatetransformation z′=f(z) is depicted in FIG. 2. The nominal focusingregion 130 and the z-position of the nominal focal plane 134 areindicated on the z-axis; the actual focusing region 140, the z′-positionof the actual focal pane 144, and the axial extent of thefocus-adjusting structure are indicated on the z′-axis. Defining a scalefactor

$\begin{matrix}{{s = {\frac{\mathbb{d}z^{\prime}}{\mathbb{d}z} = {f^{\prime}(z)}}},} & (5)\end{matrix}$the example of FIGS. 1-2 presents a constant scale factor s>1 within thefocus-adjusting structure 120, corresponding to a uniform spatialdilation. Supposing that the focus-adjusting structure is surrounded byan ambient isotropic medium with constitutive parameters ∈^(ij)=∈δ^(ij),μ^(ij)=μδ^(ij), the constitutive parameters of the transformation mediumare obtained from equations (1) and (2) and are given by (in a basiswith unit vectors 161, 162, and 160, respectively, in FIG. 1)

$\begin{matrix}{{\overset{\sim}{ɛ} = {\begin{pmatrix}s^{- 1} & 0 & 0 \\0 & s^{- 1} & 0 \\0 & 0 & s\end{pmatrix}ɛ}},{\overset{\sim}{\mu} = {\begin{pmatrix}s^{- 1} & 0 & 0 \\0 & s^{- 1} & 0 \\0 & 0 & s\end{pmatrix}{\mu.}}}} & (6)\end{matrix}$Thus, the uniform spatial dilation of FIGS. 1-2 corresponds to atransformation medium that is a uniform uniaxial medium.

In some embodiments, the focus-adjusting structure includes atransformation medium that provides a non-uniform spatial dilation. Anexample is depicted in FIG. 3 and the corresponding coordinatetransformation z′=f(z) is depicted in FIG. 4. In FIG. 3, as in FIG. 1, afocusing structure 110 provides a nominal focusing region 130 having anominal depth of field 132, and the focus-adjusting structure 120provides an actual focusing region 140 having an actual depth of field142, where the actual depth of field is an extended depth of fieldgreater than the nominal depth of field. In contrast to FIG. 1, however,the embodiment of FIG. 3 provides an actual focal plane 144 coincidentwith the nominal focal plane 134; moreover, after exiting thefocus-adjusting structure, the actual rays 103 and the nominal rays 104propagate identically (implying that an optical path length through thefocus-adjusting structure is equal to a nominal optical path lengthwhere the focus-adjusting structure is replaced by an ambient medium).These attributes are demonstrated in FIG. 4, where the dashed line z′=zintersects z′=f(z) at the position of the focal plane, and at theendpoints of the focus-adjusting structure. The example of FIGS. 3-4presents a non-uniform scale factor s (the slope of the mapping functionz′=f(z)); indeed, the scale factor in this example is variously lessthan unity (corresponding to a local spatial compression) and greaterthan unity (corresponding to a local spatial dilation). The constitutiverelations are again given by equations (6), where s is variable in theaxial direction, and the transformation medium is a non-uniform uniaxialmedium.

More generally, embodiments of a focus-adjusting structure, operable toprovide an extended depth of focus for output electromagnetic energygreater than the nominal depth of focus, may comprise a transformationmedium, the transformation medium corresponding to a coordinatetransformation that maps a nominal focusing region (with a nominal depthof focus) to an actual focusing region (with an actual depth of focusgreater than the nominal depth of focus); and the constitutive relationsof this transformation medium may be implemented with anartificially-structured material (such as a metamaterial), as describedpreviously. In some embodiments, the coordinate transformation from thenominal focusing region to the actual focusing region includes a spatialdilation along an axial direction of the nominal focusing region, and ascale factor of the spatial dilation (within the actual focusing region)may correspond to a ratio of the actual depth of focus to the nominaldepth of focus. This is consistent with FIGS. 2 and 4, where the slopetriangle 200, indicating the scale factor on the focal plane, is similaror substantially similar to a triangle with a base 132 and a height 142.Just as the axial direction can vary along a transverse extent of thenominal focusing region, the direction of the spatial dilation can varyas well. Thus, for example, a substantially cylindrically- orspherically-curved actual focusing region may be a (uniform ornon-uniform) radial dilation of a substantially cylindrically- orspherically-curved nominal focusing region; a substantiallyellipsoidally-curved actual focusing region may be a (uniform ornon-uniform) confocal dilation of a substantially ellipsoidally-curvednominal focusing region; etc.

In some embodiments, where the focusing structure defines an f-numberf/# as discussed previously, the influence of the focus-adjustingstructure provides a modified relationship (as compared to equation (4))between the f-number, the nominal depth of focus, and the resolutionlength l. Namely, some embodiments provide the relation

$\begin{matrix}{{{f/{\left. \# \right.\sim\;\frac{1}{s}}}\frac{d_{A}}{\ell}},} & (7)\end{matrix}$where d_(A) is the actual depth of focus and s is a scale factor for aspatial dilation applied along the axial direction. The f-number is heredefined independently of the focus-adjusting structure: it is a ratio ofthe nominal focal path length to aperture diameter for the focusingstructure (however, some embodiments provide an actual focal path lengthequal or substantially equal to the nominal focal path length,notwithstanding that the actual focal distance may be substantiallydifferent than the nominal focal distance). Combining equations (4) and(7) recovers the relation d_(A)˜sd_(N) discussed in the precedingparagraph.

The focus-adjusting structure 120 is depicted in FIGS. 1 and 3 as aplanar slab, but this is a schematic illustration and is not intended tobe limiting. In various embodiments the focus-adjusting structure can bea cylindrically-, spherically-, or ellipsoidally-curved slab, or anyother slab- or non-slab-like structure configured to receive the outputelectromagnetic energy and provide an extended depth of focus greaterthan the nominal depth of focus. In some embodiments, such as thatdepicted in FIG. 5, the focus-adjusting structure 120 and the focusingstructure 110 may have a spatial separation, defining an intermediateregion 500 between the structures; in other embodiments, such as thatdepicted in FIG. 6, the focus-adjusting structure 120 and the focusingstructure 110 may define a composite or contiguous unit. Embodiments maydefine an input surface region (e.g. region 510 in FIGS. 5 and 6) as asurface region of the focus-adjusting structure that receives outputelectromagnetic radiation from the focusing structure, and this inputsurface region may be substantially nonreflective of the outputelectromagnetic radiation. For example, where the focus-adjustingstructure is a transformation medium, equations (1) and (2) generallyprovide a medium that is substantially nonreflective. More generally,the input surface region may be substantially nonreflective by virtue ofa substantial impedance-matching to the adjacent medium. When thefocusing structure and a focus-adjusting structure that are spatiallyseparated, the adjacent medium corresponds to the intermediate region(e.g. 500 in FIG. 5). When the focusing structure and a focus-adjustingstructure are adjacent, the adjacent medium corresponds to an outputsurface region 600 (e.g. as depicted in FIG. 6) of the focusingstructure.

With impedance-matching, a wave impedance of the input surface region issubstantially equal to a wave impedance of the adjacent medium. The waveimpedance of an isotropic medium is

$\begin{matrix}{Z_{0} = \sqrt{\frac{\mu}{ɛ}}} & (8)\end{matrix}$while the wave impedance of a generally anisotropic medium is a tensorquantity, e.g. as defined in L. M. Barkovskii and G. N. Borzdov, “Theimpedance tensor for electromagnetic waves in anisotropic media,” J.Appl. Spect. 20, 836 (1974) (herein incorporated by reference). In someembodiments an impedance-matching is a substantial matching of everymatrix element of the wave impedance tensor (i.e. to provide asubstantially nonreflective interface for all incident polarizations);in other embodiments an impedance-matching is a substantial matching ofonly selected matrix elements of the wave impedance tensor (e.g. toprovide a substantially nonreflective interface for a selectedpolarization only). In some embodiments, the adjacent medium defines apermittivity ∈₁ and a permeability μ₁, where either or both parametersmay be substantially unity or substantially non-unity; the input surfaceregion defines a permittivity ∈₁ and a permeability μ₂, where either orboth parameters may be substantially unity or substantially non-unity;and the impedance-matching condition implies

$\begin{matrix}{\frac{ɛ_{2}}{ɛ_{1}} \cong \frac{\mu_{2}}{\mu_{1}}} & (9)\end{matrix}$where ∈₂ and μ₂ may be tensor quantities. Defining a surface normaldirection and a surface parallel direction (e.g. depicted as elements521 and 522, respectively, in FIGS. 5 and 6), some embodiments provide ainput surface region that defines: a surface normal permittivity ∈₂ ^(⊥)corresponding to the surface normal direction and a surface parallelpermittivity ∈₂ ^(∥) corresponding to the surface parallel direction;and/or a surface normal permeability μ₂ ^(⊥) corresponding to thesurface normal direction and a surface parallel permeability μ₂ ^(∥)corresponding to the surface parallel direction; and theimpedance-matching condition may imply (in addition to equation (9)) oneor both of the following conditions:

$\begin{matrix}{{\frac{ɛ_{2}^{\bot}}{ɛ_{1}} \cong \frac{ɛ_{1}}{ɛ_{2}^{||}}},{\frac{\mu_{2}^{\bot}}{\mu_{1}} \cong {\frac{\mu_{1}}{\mu_{2}^{||}}.}}} & (10)\end{matrix}$Where the input surface region is a curved surface region (e.g. as inFIG. 6), the surface normal direction and the surface parallel directioncan vary with position along the input surface region.

Some embodiments provide one or more electromagnetic sensors positionedwithin the actual focusing region of the focusing structure. In general,electromagnetic sensors, such as those depicted FIG. 1 and in otherembodiments, are electromagnetic devices having a detectable response toreceived or absorbed electromagnetic energy. Electromagnetic sensors caninclude antennas (such as wire/loop antennas, horn antennas, reflectorantennas, patch antennas, phased array antennas, etc.), solid-statephotodetectors (such as photodiodes, CCDs, and photoresistors), vacuumphotodetectors (such as phototubes and photomultipliers) chemicalphotodetectors (such as photographic emulsions), cryogenicphotodetectors (such as bolometers), photoluminescent detectors (such asphosphor powders or fluorescent dyes/markers), MEMS detectors (such asmicrocantilever arrays with electromagnetically responsive materials orelements) or any other devices operable to detect and/or transduceelectromagnetic energy. Some embodiments include a plurality ofelectromagnetic sensors positioned within the actual focusing region. Afirst example is a multiplet of sensors operable at a correspondingmultiplet of wavelengths or wavelength bands, i.e. a first sensoroperable at a first wavelength/wavelength band, a second sensor operableat a second wavelength/wavelength band, etc. A second example is a focalplane array of sensors or sensor multiplets (e.g. a Bayer or Foveonsensor). A third example is a phased array of antennas. The plurality ofsensors can be axially distributed (as in FIG. 1); for example, theextended depth of focus may admit a plurality of parallel focal planesensor arrays. As discussed earlier, a transverse extent of thesensitive area of a sensor (or sensor multiplet) can provide aresolution length in the transverse direction (e.g. 136 in FIG. 1), andmay bear a relation to the depth of focus (as in equations (4) and (7)).

In some embodiments the focus-adjusting structure provides an extendeddepth of focus for output electromagnetic energy at a selectedfrequency/frequency band and/or a selected polarization. The selectedfrequency or frequency band may be selected from a range that includesradio frequencies, microwave frequencies, millimeter- orsubmillimeter-wave frequencies, THz-wave frequencies, opticalfrequencies (e.g. variously corresponding to soft x-rays, extremeultraviolet, ultraviolet, visible, near-infrared, infrared, or farinfrared light), etc. The selected polarization may be a particular TEpolarization (e.g. where the electric field is in a particular directiontransverse to the axial direction, as with s-polarized electromagneticenergy), a particular TM polarization (e.g. where the magnetic field isin a particular direction transverse to the axial direction, as withp-polarized electromagnetic energy), a circular polarization, etc.(other embodiments provide an extended depth of focus for outputelectromagnetic energy that is substantially the same extended depth offocus for any polarization, e.g. for unpolarized electromagneticenergy).

In other embodiments the focus-adjusting structure provides a firstextended depth of focus for output electromagnetic energy at a firstfrequency and a second extended depth of focus for outputelectromagnetic energy at a second frequency, where the second extendeddepth of focus may be different than or substantially equal to the firstextended depth of focus. For embodiments that recite first and secondfrequencies, the first and second frequencies may be selected from thefrequency categories in the preceding paragraph. Moreover, for theseembodiments, the recitation of first and second frequencies maygenerally be replaced by a recitation of first and second frequencybands, again selected from the above frequency categories. Theseembodiments providing a focus-adjusting structure operable at first andsecond frequencies may include a transformation medium having anadjustable response to electromagnetic radiation. For example, thetransformation medium may have a response to electromagnetic radiationthat is adjustable (e.g. in response to an external input or controlsignal) between a first response and a second response, the firstresponse providing the first extended depth of focus for outputelectromagnetic energy at the first frequency, and the second responseproviding the second extended depth of focus for output electromagneticenergy at the second frequency. A transformation medium with anadjustable electromagnetic response may be implemented with variablemetamaterials, e.g. as described in R. A. Hyde et al, supra. Otherembodiments of a focus-adjusting structure operable at first and secondfrequencies may include transformation medium having afrequency-dependent response to electromagnetic radiation, correspondingto frequency-dependent constitutive parameters. For example, thefrequency-dependent response at a first frequency may provide a firstextended depth of focus for output electromagnetic energy at the firstfrequency, and the frequency-dependent response at a second frequencymay provide second extended depth of focus for output electromagneticenergy at the second frequency. A transformation medium having afrequency-dependent response to electromagnetic radiation can beimplemented with artificially-structured materials such asmetamaterials; for example, a first set of metamaterial elements havinga response at the first frequency may be interleaved with a second setof metamaterial elements having a response at the second frequency.

In some embodiments the focusing structure provides a first nominaldepth of focus for output electromagnetic energy at a first frequencyand a second nominal depth of focus for output electromagnetic energy ata second frequency, where the second nominal depth of focus may bedifferent than or substantially equal to the first nominal depth offocus. Examples of focusing structures providing different first andsecond nominal depths of focus include: dichroic lenses or mirrors,frequency-selective surfaces, diffractive gratings; metamaterials havinga frequency-dependent response; etc.; or, generally, any focusingstructure having a chromatic dispersion or aberration. A focusingstructure that provides first and second nominal depths of focus foroutput electromagnetic energy at first and second frequencies can becombined with a focus-adjusting structure that provides first and secondextended depths of focus for output electromagnetic energy at the firstand second frequencies. A particular embodiment provides different firstand second nominal depths of focus but substantially equal first andsecond extended depths of focus (thus, for example, compensating for achromatic aberration of the focusing structure).

An illustrative embodiment is depicted as a process flow diagram in FIG.7. Flow 700 optionally includes operation 710—deflecting anelectromagnetic wave, whereby the electromagnetic wave converges towardsa nominal focusing region having a nominal axial dimension. For example,a focusing structure, such as that depicted as element 110 in FIGS. 1and 3, deflects input electromagnetic energy 102, whereby outputelectromagnetic energy 103 converges towards a nominal focusing region130. Operation 710 optionally includes sub-operation 712—deflecting afirst component of the electromagnetic wave at a first frequency,whereby the first component converges toward a first nominal focusingsubregion within the nominal focusing region, the first nominal focusingsubregion having a first nominal axial subdimension—and sub-operation714—deflecting a second component of the electromagnetic wave at asecond frequency, whereby the second electromagnetic wave convergestoward a second nominal focusing subregion within the nominal focusingregion, the second nominal focusing subregion having a second nominalaxial subdimension. For example, a focusing structure may provide afirst nominal depth of focus for output electromagnetic energy at afirst frequency and a second nominal depth of focus for outputelectromagnetic energy at a second frequency (e.g. where the focusingstructure has a chromatic dispersion or aberration). Flow 700 includesoperation 720—substantially-nonreflectively receiving theelectromagnetic wave that converges towards the nominal focusing region.For example, a focus-adjusting structure, such as that depicted aselement 120 in FIGS. 5 and 6, may include an input surface region 510that is substantially nonreflective of input electromagnetic energyincident upon the input surface region from an adjacent region (e.g.where the input surface region is substantially impedance-matched to theadjacent region). Flow 700 further includes operation 730—spatiallydilating the electromagnetic wave along a direction corresponding to thenominal axial dimension, thereby providing an actual focusing regionhaving an actual axial dimension greater than the nominal axialdimension. For example, a focus-adjusting structure, such as thatdepicted as element 120 in FIGS. 1 and 3, may receive outputelectromagnetic energy 103 and influence the propagation thereof,whereby the output electromagnetic energy converges towards an actualfocusing region 140 instead of a nominal focusing region 130; and thefocus-adjusting structure may include a transformation mediumcorresponding to a coordinate transformation that includes a spatialdilation, e.g. as depicted in FIGS. 2 and 4. Operation 730 optionallyincludes sub-operation 732—spatially dilating a first component of theelectromagnetic wave at a first frequency, thereby providing a firstactual focusing subregion within the actual focusing region, the firstactual focusing subregion having a first actual axial subdimensiongreater than the nominal axial dimension—and sub-operation 734—spatiallydilating a second electromagnetic wave at a second frequency, therebyproviding a second actual focusing subregion within the actual focusingregion, the second actual focusing subregion having a second actualaxial subdimension greater than the nominal axial dimension. Forexample, a focus-adjusting structure may provide a first extended depthof focus for output electromagnetic energy at a first frequency and asecond extended depth of focus for output electromagnetic energy at asecond frequency, where the second extended depth of focus may bedifferent than or substantially equal to the first extended depth offocus; and the focus-adjusting structure operable at first and secondfrequencies may include a transformation medium having an adjustableresponse to electromagnetic radiation, or a transformation medium havinga frequency-dependent response to electromagnetic radiation. Flow 700optionally further includes operation 740—sensing the electromagneticwave at one or more locations within the actual focusing region. Forexample, one or more electromagnetic sensors, such as those depicted aselements 150 in FIG. 1, may be positioned within an actual focusingregion 140 to detect/receive/absorb the output electromagnetic energy103.

Another illustrative embodiment is depicted as a process flow diagram inFIG. 8. Flow 800 includes operation 810—identifying a nominal focusingregion for a focusing structure, the nominal focusing region having anominal axial dimension. For example, a nominal focusing region isdepicted as region 130 in FIGS. 1 and 3. The nominal focusing region maycorrespond to a region centered on a nominal focal plane (or non-planarnominal focal surface) having a thickness corresponding to a nominaldepth of focus. Operation 810 optionally includes sub-operation 812—forelectromagnetic waves at a first frequency, identifying a first nominalfocusing subregion within the nominal focusing region for the focusingstructure, the first nominal focusing subregion having a first nominalaxial subdimension—and sub-operation 814—for electromagnetic waves at asecond frequency, identifying a second nominal focusing subregion withinthe nominal focusing region for the focusing structure, the secondnominal focusing subregion having a second nominal axial subdimension.For example, the focusing structure may have a chromatic dispersion oraberration, whereby the nominal focusing region depends upon thefrequency of the input electromagnetic energy. Flow 800 further includesoperation 820—determining electromagnetic parameters for a spatialregion containing the nominal focusing region, the electromagneticparameters providing an actual focusing region having an actual axialdimension greater than the nominal axial dimension; where theelectromagnetic parameters include (1) axial electromagnetic parametersand (2) transverse electromagnetic parameters that inversely correspondto the axial electromagnetic parameters. For example, the spatial regioncan be a volume that encloses a focus-adjusting structure, such as thatdepicted as element 120 in FIGS. 1 and 3, and the determinedelectromagnetic parameters are the electromagnetic parameters of thefocus-adjusting structure. The focus-adjusting structure may include atransformation medium, where the determined electromagnetic parameterssatisfy or substantially satisfy equations (1) and (2), as describedabove; or, the determined electromagnetic parameters may be reducedparameters (as discussed earlier) where the corresponding non-reducedparameters satisfy equations (1) and (2). In some embodiments, thedetermining of the electromagnetic parameters includes: determining acoordinate transformation (such as those depicted in FIGS. 2 and 4);then determining electromagnetic parameters for a correspondingtransformation medium (e.g. with equations (1) and (2)); then,optionally, reducing the electromagnetic parameters (e.g. to at leastpartially substitute a magnetic response for an electromagneticresponse, or vice versa, as discussed above). Operation 820 optionallyincludes sub-operation 822—for electromagnetic waves at a firstfrequency, determining a first subset of the electromagnetic parametersproviding a first actual focusing subregion within the actual focusingregion, the first actual focusing subregion having a first actual axialsubdimension greater than the nominal axial dimension—and sub-operation824—for electromagnetic waves at a second frequency, determining asecond subset of the electromagnetic parameters providing a secondactual focusing subregion within the actual focusing region, the secondactual focusing subregion having a second actual axial subdimensiongreater than the nominal axial dimension. For example, the determinedelectromagnetic parameters may be the electromagnetic parameters of afocus-adjusting structure having a first extended depth of focus foroutput electromagnetic energy at a first frequency and a second extendeddepth of focus for output electromagnetic energy at a second frequency.The focus-adjusting structure may include a transformation medium havingan adjustable response to electromagnetic radiation, e.g. adjustablebetween a first response, corresponding to the first subset of theelectromagnetic parameters, and a second response, corresponding to thesecond subset of the electromagnetic parameters. Or, the focus-adjustingstructure may include a transformation medium having afrequency-dependent response to electromagnetic radiation, correspondingto frequency-dependent constitutive parameters, so that the first andsecond subsets of the electromagnetic parameters are values of thefrequency-dependent constitutive parameters at the first and secondfrequencies, respectively. Flow 800 optionally further includesoperation 830—selecting one or more positions of one or moreelectromagnetic sensors within the spatial region. For example,electromagnetic sensors may be positioned in a phased array, a focalplane array, an axially-distributed arrangement, etc. Flow 800optionally further includes operation 840—configuring anartificially-structured material having an effective electromagneticresponse that corresponds to the electromagnetic parameters in thespatial region. For example, the configuring may include configuring thestructure(s) and/or the materials that compose a photonic crystal or ametamaterial. Operation 840 optionally includes determining anarrangement of a plurality of electromagnetically responsive elementshaving a plurality of individual responses, the plurality of individualresponses composing the effective electromagnetic response. For example,the determining may include determining the positions, orientations, andindividual response parameters (spatial dimensions, resonantfrequencies, linewidths, etc.) of a plurality of metamaterial elementssuch as split-ring resonators, wire or nanowire pairs, etc. Operation840 optionally includes configuring at least oneelectromagnetically-responsive structure to arrange a plurality ofdistributed electromagnetic responses, the plurality of distributedelectromagnetic responses composing the effective electromagneticresponse. For example, the configuring may include configuring thedistribution of loads and interconnections on a transmission linenetwork, configuring an arrangement of layers in a layered metamaterial,configuring a pattern of etching or deposition (as with a nano-fishnetstructure), etc.

With reference now to FIG. 9, an illustrative embodiment is depicted asa system block diagram. The system 900 includes a focusing unit 910optionally coupled to a controller unit 940. The focusing unit 910 mayinclude a focusing structure such as that depicted as element 110 inFIGS. 1 and 3. The focusing structure may be a variable or adaptivefocusing structure, such as an electro-optic lens, a liquid orliquid-crystal lens, a mechanically-adjustable lens assembly, a variablemetamaterial lens, or any other variable or adaptive focusing structureresponsive to one or more control inputs to vary or adapt one or morefocusing characteristics (focal length, aperture size, nominal depth offocus, operating frequency/frequency band, operating polarization, etc.)of the focusing structure; and the controller unit 940 may includecontrol circuitry that provides one or more control inputs to thevariable or adaptive focusing structure. The system 900 further includesa focus-adjusting unit 920 coupled to the controller unit 940. Thefocusing-adjusting unit 920 may include a focus-adjusting structure suchas that depicted as element 120 in FIGS. 1 and 3. The focus-adjustingstructure may be a variable focus-adjusting structure, such as avariable metamaterial responsive to one or more control inputs to varyone or more focus-adjusting characteristics (extended depth of focus,operating frequency/frequency band, operating polarization, effectivecoordinate transformation for a transformation medium, etc.); and thecontroller unit 940 may include control circuitry that provides one ormore control inputs to the variable focus-adjusting structure. Thecontroller unit 940 may include circuitry for coordinating orsynchronizing the operation of the focusing unit 910 and thefocus-adjusting unit 920; for example, the controller unit 940 may varyone or more focusing characteristics of a focusing structure (e.g. varyan operating frequency of the focusing structure from a first frequencyto a second frequency) and correspondingly vary one or morefocus-adjusting characteristics of a focus-adjusting structure (e.g.vary an operating frequency of the focus-adjusting structure from afirst frequency to a second frequency). The system 900 optionallyfurther includes a sensing unit 930 that may include one or moresensors, such as those depicted as elements 150 in FIG. 1, andassociated circuitry such as receiver circuitry, detector circuitry,and/or signal processing circuitry. The sensing unit 930 is optionallycoupled to the controller unit 940, and in some embodiments thecontroller unit 940 includes circuitry responsive to sensor data (fromthe sensor unit 930) to vary the focusing characteristics of a focusingstructure and/or vary the focus-adjusting characteristics of afocus-adjusting structure. As a first example, the controller unit 940may include circuitry responsive to the sensor data to identify afrequency of received energy, adjust the focusing unit 910 to anoperating frequency substantially equal to the frequency of receivedenergy, and/or adjust the focus-adjusting unit 920 to an operatingfrequency substantially equal to the frequency of received energy. As asecond example, the controller unit may include circuitry responsive tothe sensor data to identify an actual focusing region (e.g. depicted asregion 140 in FIGS. 1 and 3) and vary the actual focusing region byvarying the focusing characteristics of a focusing structure and/orvarying the focus-adjusting characteristics of a focus-adjustingstructure.

The foregoing detailed description has set forth various embodiments ofthe devices and/or processes via the use of block diagrams, flowcharts,and/or examples. Insofar as such block diagrams, flowcharts, and/orexamples contain one or more functions and/or operations, it will beunderstood by those within the art that each function and/or operationwithin such block diagrams, flowcharts, or examples can be implemented,individually and/or collectively, by a wide range of hardware, software,firmware, or virtually any combination thereof. In one embodiment,several portions of the subject matter described herein may beimplemented via Application Specific Integrated Circuits (ASICs), FieldProgrammable Gate Arrays (FPGAs), digital signal processors (DSPs), orother integrated formats. However, those skilled in the art willrecognize that some aspects of the embodiments disclosed herein, inwhole or in part, can be equivalently implemented in integratedcircuits, as one or more computer programs running on one or morecomputers (e.g., as one or more programs running on one or more computersystems), as one or more programs running on one or more processors(e.g., as one or more programs running on one or more microprocessors),as firmware, or as virtually any combination thereof, and that designingthe circuitry and/or writing the code for the software and or firmwarewould be well within the skill of one of skill in the art in light ofthis disclosure. In addition, those skilled in the art will appreciatethat the mechanisms of the subject matter described herein are capableof being distributed as a program product in a variety of forms, andthat an illustrative embodiment of the subject matter described hereinapplies regardless of the particular type of signal bearing medium usedto actually carry out the distribution. Examples of a signal bearingmedium include, but are not limited to, the following: a recordable typemedium such as a floppy disk, a hard disk drive, a Compact Disc (CD), aDigital Video Disk (DVD), a digital tape, a computer memory, etc.; and atransmission type medium such as a digital and/or an analogcommunication medium (e.g., a fiber optic cable, a waveguide, a wiredcommunications link, a wireless communication link, etc.).

In a general sense, those skilled in the art will recognize that thevarious aspects described herein which can be implemented, individuallyand/or collectively, by a wide range of hardware, software, firmware, orany combination thereof can be viewed as being composed of various typesof “electrical circuitry.” Consequently, as used herein “electricalcircuitry” includes, but is not limited to, electrical circuitry havingat least one discrete electrical circuit, electrical circuitry having atleast one integrated circuit, electrical circuitry having at least oneapplication specific integrated circuit, electrical circuitry forming ageneral purpose computing device configured by a computer program (e.g.,a general purpose computer configured by a computer program which atleast partially carries out processes and/or devices described herein,or a microprocessor configured bad a computer program which at leastpartially carries out processes and/or devices described herein),electrical circuitry forming a memory device (e.g., forms of randomaccess memory), and/or electrical circuitry forming a communicationsdevice (e.g., a modem, communications switch, or optical-electricalequipment). Those having skill in the art will recognize that thesubject matter described herein may be implemented in an analog ordigital fashion or some combination thereof.

All of the above U.S. patents, U.S. patent application publications,U.S. patent applications, foreign patents, foreign patent applicationsand non-patent publications referred to in this specification and/orlisted in any Application Data Sheet, are incorporated herein byreference, to the extent not inconsistent herewith.

One skilled in the art will recognize that the herein describedcomponents (e.g., steps), devices, and objects and the discussionaccompanying them are used as examples for the sake of conceptualclarity and that various configuration modifications are within theskill of those in the art. Consequently, as used herein, the specificexemplars set forth and the accompanying discussion are intended to berepresentative of their more general classes. In general, use of anyspecific exemplar herein is also intended to be representative of itsclass, and the non-inclusion of such specific components (e.g., steps),devices, and objects herein should not be taken as indicating thatlimitation is desired.

With respect to the use of substantially any plural and/or singularterms herein, those having skill in the art can translate from theplural to the singular and/or from the singular to the plural as isappropriate to the context and/or application. The varioussingular/plural permutations are not expressly set forth herein for sakeof clarity.

While particular aspects of the present subject matter described hereinhave been shown and described, it will be apparent to those skilled inthe art that, based upon the teachings herein, changes and modificationsmay be made without departing from the subject matter described hereinand its broader aspects and, therefore, the appended claims are toencompass within their scope all such changes and modifications as arewithin the true spirit and scope of the subject matter described herein.Furthermore, it is to be understood that the invention is defined by theappended claims. It will be understood by those within the art that, ingeneral, terms used herein, and especially in the appended claims (e.g.,bodies of the appended claims) are generally intended as “open” terms(e.g., the term “including” should be interpreted as “including but notlimited to,” the term “having” should be interpreted as “having atleast,” the term “includes” should be interpreted as “includes but isnot limited to,” etc.). It will be further understood by those withinthe art that if a specific number of an introduced claim recitation isintended, such an intent will be explicitly recited in the claim, and inthe absence of such recitation no such intent is present. For example,as an aid to understanding, the following appended claims may containusage of the introductor, phrases “at least one” and “one or more” tointroduce claim recitations. However, the use of such phrases should notbe construed to imply that the introduction of a claim recitation by theindefinite articles “a” or “an” limits any particular claim containingsuch introduced claim recitation to inventions containing only one suchrecitation, even when the same claim includes the introductory phrases“one or more” or “at least one” and indefinite articles such as “a” or“an” (e.g., “a” and/or “an” should typically be interpreted to mean “atleast one” or “one or more”); the same holds true for the use ofdefinite articles used to introduce claim recitations. In addition, evenif a specific number of an introduced claim recitation is explicitlyrecited, those skilled in the art will recognize that such recitationshould typically be interpreted to mean at least the recited number(e.g., the bare recitation of “two recitations,” without othermodifiers, typically means at least two recitations, or two or morerecitations). Furthermore, in those instances where a conventionanalogous to “at least one of A, B, and C, etc.” is used, in generalsuch a construction is intended in the sense one having skill in the artwould understand the convention (e.g., “a system having at least one ofA, B, and C” would include but not be limited to systems that have Aalone, B alone, C alone, A and B together, A and C together, B and Ctogether, and/or A, B, and C together, etc.). In those instances where aconvention analogous to “at least one of A, B, or C, etc.” is used, ingeneral such a construction is intended in the sense one having skill inthe art would understand the convention (e.g., “a system having at leastone of A, B, or C” would include but not be limited to systems that haveA alone, B alone, C alone, A and B together, A and C together, B and Ctogether, and/or A, B, and C together, etc.). It will be furtherunderstood by those within the art that virtually any disjunctive wordand/or phrase presenting two or more alternative terms, whether in thedescription, claims, or drawings, should be understood to contemplatethe possibilities of including one of the terms, either of the terms, orboth terms. For example, the phrase “A or B” will be understood toinclude the possibilities of “A” or “B” or “A and B.”

With respect to the appended claims, those skilled in the art willappreciate that recited operations therein may generally be performed inany order. Examples of such alternate orderings may include overlapping,interleaved, interrupted, reordered, incremental, preparatory,supplemental, simultaneous, reverse, or other variant orderings, unlesscontext dictates otherwise. With respect to context, even terms like“responsive to,” “related to,” or other past-tense adjectives aregenerally not intended to exclude such variants, unless context dictatesotherwise.

While various aspects and embodiments have been disclosed herein, otheraspects and embodiments will be apparent to those skilled in the art.The various aspects and embodiments disclosed herein are for purposes ofillustration and are not intended to be limiting, with the true scopeand spirit being indicated by the following claims.

1. A method, comprising: identifying a nominal focusing region for afocusing structure, the nominal focusing region having a nominal axialdimension; determining electromagnetic parameters for a spatial regioncontaining the nominal focusing region, the electromagnetic parametersproviding an actual focusing region having an actual axial dimensiongreater than the nominal axial dimension; where the electromagneticparameters include: axial electromagnetic parameters; and transverseelectromagnetic parameters that inversely correspond to the axialelectromagnetic parameters; and selecting one or more positions of oneor more electromagnetic sensors within the spatial region.
 2. The methodof claim 1, wherein the one or more positions of the one or moreelectromagnetic sensors is a plurality of positions for a plurality ofelectromagnetic sensors.
 3. The method of claim 2, wherein the pluralityof electromagnetic sensors composes a sensor phased array.
 4. The methodof claim 2, wherein the plurality of positions is an axially distributedplurality of positions.
 5. A method, comprising: identifying a nominalfocusing region for a focusing structure, the nominal focusing regionhaving a nominal axial dimension; and determining electromagneticparameters for a spatial region containing the nominal focusing region,the electromagnetic parameters providing an actual focusing regionhaving an actual axial dimension greater than the nominal axialdimension; where the electromagnetic parameters include: axialelectromagnetic parameters; and transverse electromagnetic parametersthat inversely correspond to the axial electromagnetic parameters;wherein the electromagnetic parameters define a spatially-varying indexof refraction within the spatial region.
 6. A method, comprising:identifying a nominal focusing region for a focusing structure, thenominal focusing region having a nominal axial dimension; anddetermining electromagnetic parameters for a spatial region containingthe nominal focusing region, the electromagnetic parameters providing anactual focusing region having an actual axial dimension greater than thenominal axial dimension; where the electromagnetic parameters include:axial electromagnetic parameters; and transverse electromagneticparameters that inversely correspond to the axial electromagneticparameters; wherein the electromagnetic parameters define an index ofrefraction having an absolute value less than one within at least aportion of the spatial region.
 7. A method, comprising: identifying anominal focusing region for a focusing structure, the nominal focusingregion having a nominal axial dimension; and determining electromagneticparameters for a spatial region containing the nominal focusing region,the electromagnetic parameters providing an actual focusing regionhaving an actual axial dimension greater than the nominal axialdimension; where the electromagnetic parameters include: axialelectromagnetic parameters; and transverse electromagnetic parametersthat inversely correspond to the axial electromagnetic parameters;wherein the axial electromagnetic parameters include an axialpermittivity.
 8. The method of claim 7, wherein the transverseelectromagnetic parameters include a transverse permittivity that issubstantially a multiplicative inverse of the axial permittivity.
 9. Themethod of claim 7, wherein the axial electromagnetic parameters includean axial permeability.
 10. The method of claim 9, wherein the transverseelectromagnetic parameters include a transverse permeability that issubstantially a multiplicative inverse of the axial permeability. 11.The method of claim 9, wherein the axial permittivity is substantiallyequal to the axial permeability.
 12. The method of claim 11, wherein thetransverse electromagnetic parameters include a transverse permeabilitythat is substantially a multiplicative inverse of the axialpermeability.
 13. A method, comprising: identifying a nominal focusingregion for a focusing structure, the nominal focusing region having anominal axial dimension; and determining electromagnetic parameters fora spatial region containing the nominal focusing region, theelectromagnetic parameters providing an actual focusing region having anactual axial dimension greater than the nominal axial dimension; wherethe electromagnetic parameters include: axial electromagneticparameters; and transverse electromagnetic parameters that inverselycorrespond to the axial electromagnetic parameters; wherein the focusingstructure and the spatial region have a physical separation, thephysical separation defines an intermediate region between the focusingstructure and the spatial region, the spatial region includes an inputsurface region that is substantially nonreflective of electromagneticenergy incident upon the input surface region from the intermediateregion, and the input surface region is substantially impedance-matchedto the intermediate region.
 14. The method of claim 13, wherein a waveimpedance of the intermediate region is substantially equal to a waveimpedance of the input surface region.
 15. The method of claim 14,wherein the intermediate region defines a first permittivity and a firstpermeability, the input surface region defines a second permittivity anda second permeability, and a ratio of the second permittivity to thefirst permittivity is substantially equal to a ratio of the secondpermeability to the first permeability.
 16. The method of claim 15,wherein the first permittivity is substantially nonunity.
 17. The methodof claim 15, wherein the first permeability is substantially nonunity.18. The method of claim 15, wherein the input surface region defines asurface normal direction and a surface parallel direction, the secondpermittivity includes a surface normal permittivity corresponding to thesurface normal direction and a surface parallel permittivitycorresponding to the surface parallel direction, and a ratio of thesurface normal permittivity to the first permittivity is substantially amultiplicative inverse of a ratio of the surface parallel permittivityto the first permittivity.
 19. The method of claim 18, wherein thesecond permeability includes a surface normal permeability correspondingto the surface normal direction and a surface parallel permeabilitycorresponding to the surface parallel direction, and a ratio of thesurface normal permeability to the first permeability is substantially amultiplicative inverse of a ratio of the surface parallel permeabilityto the first permeability.
 20. A method, comprising: identifying anominal focusing region for a focusing structure, the nominal focusingregion having a nominal axial dimension; and determining electromagneticparameters for a spatial region containing the nominal focusing region,the electromagnetic parameters providing an actual focusing regionhaving an actual axial dimension greater than the nominal axialdimension; where the electromagnetic parameters include: axialelectromagnetic parameters; and transverse electromagnetic parametersthat inversely correspond to the axial electromagnetic parameters;wherein the spatial region is immediately adjacent to the focusingstructure, the focusing structure includes an output surface region, thespatial region includes an input surface region substantially adjoiningthe output surface region, and the input surface region is substantiallynonreflective of electromagnetic energy incident upon the input surfaceregion from the output surface region.
 21. The method of claim 20,wherein the input surface region is substantially impedance-matched tothe output surface region.
 22. The method of claim 21, wherein a waveimpedance of the output surface region is substantially equal to a waveimpedance of the input surface region.
 23. The method of claim 22,wherein the output surface region defines a first permittivity and afirst permeability, the input surface region defines a secondpermittivity and a second permeability, and a ratio of the secondpermittivity to the first permittivity is substantially equal to a ratioof the second permeability to the first permeability.
 24. The method ofclaim 23, wherein the input surface region defines a surface normaldirection and a surface parallel direction, the second permittivityincludes a surface normal permittivity corresponding to the surfacenormal direction and a surface parallel permittivity corresponding tothe surface parallel direction, and a ratio of the surface normalpermittivity to the first permittivity is substantially a multiplicativeinverse of a ratio of the surface parallel permittivity to the firstpermittivity.
 25. The method of claim 24, wherein the secondpermeability includes a surface normal permeability corresponding to thesurface normal direction and a surface parallel permeabilitycorresponding to the surface parallel direction, and a ratio of thesurface normal permeability to the first permeability is substantially amultiplicative inverse of a ratio of the surface parallel permeabilityto the first permeability.
 26. A method, comprising: identifying anominal focusing region for a focusing structure, the nominal focusingregion having a nominal axial dimension; and determining electromagneticparameters for a spatial region containing the nominal focusing region,the electromagnetic parameters providing an actual focusing regionhaving an actual axial dimension greater than the nominal axialdimension; where the electromagnetic parameters include: axialelectromagnetic parameters; and transverse electromagnetic parametersthat inversely correspond to the axial electromagnetic parameters;wherein the electromagnetic parameters correspond to an effectivecoordinate transformation that maps the actual focusing region to thenominal focusing region; and wherein the effective coordinatetransformation is a spatial dilation along a direction corresponding tothe nominal axial dimension.
 27. The method of claim 26, wherein thespatial dilation is a non-uniform spatial dilation.
 28. A method,comprising: identifying a nominal focusing region for a focusingstructure, the nominal focusing region having a nominal axial dimension;and determining electromagnetic parameters for a spatial regioncontaining the nominal focusing region, the electromagnetic parametersproviding an actual focusing region having an actual axial dimensiongreater than the nominal axial dimension; where the electromagneticparameters include: axial electromagnetic parameters; and transverseelectromagnetic parameters that inversely correspond to the axialelectromagnetic parameters; and configuring an artificially-structuredmaterial having an effective electromagnetic response that correspondsto the electromagnetic parameters in the spatial region; wherein theartificially-structured material includes a metamaterial.
 29. The methodof claim 28, wherein the configuring includes: determining anarrangement of a plurality of electromagnetically responsive elementshaving a plurality of individual responses, the plurality of individualresponses composing the effective electromagnetic response.
 30. Themethod of claim 29, wherein the plurality of individual responsesincludes a plurality of individual frequency responses.
 31. The methodof claim 30, wherein the plurality of individual frequency responses isa plurality of individual frequency bandwidth responses corresponding toa plurality of individual bandwidths.
 32. The method of claim 31,wherein the plurality of electromagnetically responsive elements is aplurality of electromagnetically responsive resonators corresponding toa plurality of resonant frequencies, and each of the individualbandwidths is substantially centered around a corresponding one of theresonant frequencies.
 33. The method of claim 31, wherein the individualbandwidths are at least partially overlapping.
 34. The method of claim31, wherein the individual bandwidths define a substantially contiguouscomposite bandwidth.
 35. The method of claim 29, wherein theelectromagnetically responsive elements include discrete circuitelements.
 36. The method of claim 29, wherein the electromagneticallyresponsive elements include integrated circuit elements.
 37. The methodof claim 29, wherein the electromagnetically responsive elements includemetallic structures.
 38. The method of claim 29, wherein theelectromagnetically responsive elements include LC resonators.
 39. Themethod of claim 29, wherein the electromagnetically responsive elementsinclude plasmonic resonators.
 40. The method of claim 29, wherein theelectromagnetically responsive elements include nanostructures.
 41. Themethod of claim 29, wherein the electromagnetically responsive elementsinclude split-ring resonators.
 42. The method of claim 28, wherein theconfiguring includes: configuring at least oneelectromagnetically-responsive structure to arrange a plurality ofdistributed electromagnetic responses, the plurality of distributedelectromagnetic responses composing the effective electromagneticresponse.
 43. The method of claim 42, wherein the distributedelectromagnetic responses include distributed inductive-capacitiveresponses.
 44. The method of claim 42, wherein the at least oneelectromagnetically-responsive structure includes transmission lines.